Electronic ISBN:  9781470404123 
Product Code:  MEMO/171/811.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 171; 2004; 214 ppMSC: Primary 17; 20;
We develop the basic theory of root systems \(R\) in a real vector space \(X\) which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: The intersection of \(R\) with every finitedimensional subspace of \(X\) is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings.
ReadershipGraduate students and research mathematicians interested in infinitedimensional Lie theory.

Table of Contents

Chapters

Introduction

1. The category of sets in vector spaces

2. Finiteness conditions and bases

3. Locally finite root systems

4. Invariant inner products and the coroot system

5. Weyl groups

6. Integral bases, root bases and Dynkin diagrams

7. Weights and coweights

8. Classification

9. More on Weyl groups and automorphism groups

10. Parabolic subsets and positive systems for symmetric sets in vector spaces

11. Parabolic subsets of root systems and presentations of the root lattice and the Weyl group

12. Closed and full subsystems of finite and infinite classical root systems

13. Parabolic subsets of root systems: classification

14. Positive systems in root systems

15. Positive linear forms and facets

16. Dominant and fundamental weights

17. Gradings of root systems

18. Elementary relations and graphs in 3graded root systems


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We develop the basic theory of root systems \(R\) in a real vector space \(X\) which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: The intersection of \(R\) with every finitedimensional subspace of \(X\) is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings.
Graduate students and research mathematicians interested in infinitedimensional Lie theory.

Chapters

Introduction

1. The category of sets in vector spaces

2. Finiteness conditions and bases

3. Locally finite root systems

4. Invariant inner products and the coroot system

5. Weyl groups

6. Integral bases, root bases and Dynkin diagrams

7. Weights and coweights

8. Classification

9. More on Weyl groups and automorphism groups

10. Parabolic subsets and positive systems for symmetric sets in vector spaces

11. Parabolic subsets of root systems and presentations of the root lattice and the Weyl group

12. Closed and full subsystems of finite and infinite classical root systems

13. Parabolic subsets of root systems: classification

14. Positive systems in root systems

15. Positive linear forms and facets

16. Dominant and fundamental weights

17. Gradings of root systems

18. Elementary relations and graphs in 3graded root systems